A few days ago, the BBC introduced a series of posts in which they asked mathematicians and physicists to share their favorite equations. It’s a fun list, and the original post can be found here.
In this equation, “i” is a so-called imaginary number, defined such that . This expression, which is a special case of a more general one known as Euler’s formula, is often considered to be mathematically beautiful because it includes five of the most important mathematical constants: 0, 1, i, π, and e.
An interesting discussion arose on Twitter when Evelyn Lamb posted the following in response:
I kinda agree with her, though maybe not for exactly the same reason! For me, as a researcher who sort of balances on the line between theoretical physics and applied mathematics, “beauty” in mathematics comes from an expression that really shows you something, and leads to insights and a sense of wonder. Euler’s identity doesn’t really do it for me anymore; it contains some insight, but its main attraction is the fact that it happens to include many mathematical constants.
This led to a gauntlet of sorts being thrown down by On This Day in Math!
A fair question, and I thought I would share some of the equations that I find beautiful, taken from both math and physics.
Before I begin let me make one important point: beauty is, of course, in the eye of the beholder, and quite subjective. This list is just my personal taste, and doesn’t say anything about your favorite equation missing from the list. So don’t comment and complain about it, please.
Einstein’s field equations. In 1915, ten years after publishing his theory of special relativity, Albert Einstein released his general theory of relativity. At the center of this theory is a set of equations that relate space, time and matter, which can be condensed into a single tensor equation, the Einstein field equation:
The left side of this equation, roughly speaking, describes the curvature of space and time, while the right side of the equation describes energy, momentum, and matter. What results is a mutual back and forth between matter and the motion of spacetime; as physicist John Wheeler once said, “Space-time tells matter how to move; matter tells space-time how to curve.”
This equation is on the original BBC list, and you can read about it more there. One of the most striking aspects of the field equations is their prediction of gravitational waves, which are actively being searched for; see, for instance, the LIGO experiment. The Einstein equations are also a fundamental part of our understanding of the universe as a whole, in the field of cosmology.
Maxwell’s equations. Around 1862, James Clerk Maxwell published a preliminary form of the set of equations now named after him, which describe the relationships between electricity, magnetism, and the sources of both. In modern notation, these equations are usually written as
It may seem like cheating to use four equations as a single “equation,” but like Einstein’s equations they can be condensed to a much more compact tensor form. These equations describe how the electric field E and the magnetic field B (and related fields D and H) are related to each other and to the electric charge density ρ and the electric current J.
These equations had two immediate consequences. They showed that electricity and magnetism are really just different manifestations of a single, unified, phenomenon, now called electromagnetism. This paved the way for later attempts to unify all the fundamental forces of nature into a single grand unified theory, of which the standard model of particle physics is the strongest so far, unifying electromagnetism, the weak nuclear force and the strong nuclear force.
The other consequence of Maxwell’s work was the discovery that his completed equations support electromagnetic waves. Maxwell himself postulated, correctly, that light is an electromagnetic wave, and therefore accomplished a third unification with a single set of equations, bringing together light, electricity and magnetism as all part of a single phenomenon. The derivation of a wave equation from Maxwell’s equations is remarkably simple, and something that can be done in an undergraduate physics class.
I’ve been working on a simple, non-technical introduction to Maxwell’s equations as another blog post, as they are not difficult to understand, once one gets past the math terminology. I will return to this in the future!
Wolf equations. Here I show a little bias, and pick a pair of related equations that were discovered by my PhD advisor, Emil Wolf! The equations in equation look quite simple, and are in fact a pair of slightly modified wave equations:
In the 1950s, Wolf was working with Max Born on an comprehensive optics text, Principles of Optics, which is still considered an essential volume in the field over 60 years later. Wolf had a particular interest in optical coherence theory, which is concerned with how the random fluctuations of a light field and/or its source affect its behavior.
The quantities Γ in the above equations are known as correlation functions, and they represent average properties of a light wave measured at a pair of positions in space and time. Before Wolf’s discovery, these correlation functions were regarded as somewhat secondary quantities, which are measurable by experiment but not indicative of the underlying physics of the problem. What Wolf found, however, is that the correlation functions themselves satisfy a wave equation: roughly speaking, the randomness of light also propagates as a wave! These quantities, far from being of secondary in importance, really describe all the physics of both light propagation and the statistical fluctuations of light. This discovery really marked the beginning of optical coherence theory as a serious field of study, and Emil Wolf is considered the major founder of the field.
The derivation of the so-called “Wolf equations” (though he personally hates calling them that) is remarkably simple, almost trivial. However, they were by no means obvious. When Wolf presented his results to Max Born, Born said (and I try and quote from memory), “Wolf, you have always been such a sensible fellow, but now I fear that you have gone completely crazy!” After further discussion, however, Born became convinced of Wolf’s result.
Residue theorem. In further Twitter discussions with Evelyn Lamb, we are in agreement that complex analysis is one of the most beautiful and amazing fields of mathematics! I have mentioned the “imaginary” number i at the beginning of this post; complex analysis is concerned with functions of so-called complex numbers of the form
where x and y can be treated as coordinates in a “complex plane.” In the 1800s, Augustin-Louis Cauchy studied a special class of functions of z, known as analytic functions, and discovered a stunning number of remarkable properties of them. The culmination of this work might be said to be Cauchy’s residue theorem, which can be written in the form
The left side of this expression is the integral of an analytic function around a closed path in the complex plane; the residue theorem tells us that this integral depends only on the sum of “residues” within the closed path. “Residues” represent numbers associated with “singular” points of the function , where the function has no well-defined value. The residue theorem tells us that the value of an integral around any path is determined by the behavior of the function at a small number of singular points.
To give a rough idea of how bizarre this is: imagine that, by knowing the height of a finite number of landmarks in a city (buildings, bridges, etc.), you could determine how long it would take to drive any closed path throughout the city! In real-life, this makes no sense, but this is somewhat what the residue theorem suggests about analytic functions.
The term “imaginary number” was originally introduced by René Descartes, who used the term to be dismissive of such peculiar mathematical quantities. But the use of complex analysis has turned out to be crucially important in solving many problems in mathematics and physics; the term “imaginary” turns out to be woefully inadequate! The residue theorem, in particular, allows the solution of many problems that would otherwise be impossible.
Riemann-Zeta function. Here, I seem to be referring to a “function,” rather than an equation, but it is a particular ways of expressing that function that I find amazing. The so-called Riemann-Zeta function can be written as
The Riemann-Zeta function was introduced by Leonhard Euler but it was Bernhard Riemann who, in 1859, established the remarkable relationship above. The left side of the expression is the original definition of the zeta function, which is an infinite sum. What Riemann proved is on the right: this simple sum, which is a function of the variable s, is intimately related to the set of prime numbers!
I find this nothing short of amazing. For those who are unfamiliar, prime numbers are those that have no whole number divisors other than themselves and 1. For instance, 5 is a prime, and can only be divided by 1 and 5, while 4 is not prime, and can be divided by 1,2, or 4. The properties of prime numbers have fascinated mathematicians since ancient Greece, and today play a pivotal role in cryptography. A lot of effort has been put into finding increasingly larger prime numbers; only a few days ago, a new prime number with 22,338,618 digits was discovered!
And here, in one easily defined function, is information about the entire infinite set of primes. However, it is not so easy to get that information out: one important possibly property of the zeta function, the Riemann hypothesis, remains unproven (or disproven), though its proof would unveil key secrets about the set of primes.
This single function has been an obsession of countless mathematicians over the years, and entire books have been written about its properties, which are numerous. From simple definitions often come profound consequences.
Cantor’s inequality. Here I go off-script a little bit and use an inequality, rather than an equality (equation)! The inequality in question that I love is:
This absurdly simple expression says something profound and definite about things that we can never, even in principle, measure. It says that the the size of the infinite set of all numbers between 0 and 1, known as the continuum () is larger than the infinite set of countable numbers 1,2,3,4,… ().
That is, there are some infinities that are bigger than others!
I’ve written at length in the past about the work of Georg Cantor on the theory of transfinite numbers — see, for instance, the post linked here which begins the series. Cantor took fuzzy notions of infinite sets of numbers and built a rigorous theory about them. In fact, he found that there is an infinite hierarchy of sizes of infinite sets, and he was able to prove it in a way that is simple enough for anyone to test for themselves.
These notions of infinity may seem rather abstract and unimportant, but I have recently shown that the weirdness of infinite sets can manifest in a surprising way in optical systems. (And I’ve got a paper accepted for publication about this.) What is truly amazing about this work, however, is what I already said: we can use mathematics to make rigorous, consistent, and provable statements about things that we can never observe or experience.
So, this is my list! Looking at it, I see that I find beautiful those equations which are simple in form but reveal profound truths about nature and mathematics.
Let me know what you think the most beautiful equations are, and why, in the comments!